Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy

DOI10.1090/S0002-9947-2013-05861-8zbMATH Open1282.35143arXiv1201.2206MaRDI QIDQ2855935

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Publication date: 23 October 2013

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Abstract: We investigate $L^1(R^2)o{L}^{i}nfty(R^2)$ dispersive estimates for the Schr"odinger operator $H=-Delta+V$ when there are obstructions, resonances or an eigenvalue, at zero energy. In particular, we show that the existence of an s-wave resonance at zero energy does not destroy the $t^{-1}$ decay rate. We also show that if there is a p-wave resonance or an eigenvalue at zero energy then there is a time dependent operator $F_t$ satisfying $|F_t{|}_{L^{1}o{L}^{i}nfty}lesssim1$ such that ${\displaystyle }$|e^{itH}P_{ac}-F_t|_{L^1 o L^infty} lesssim |t|^{-1}, ext{for} |t|>1.${\displaystyle }$ We also establish a weighted dispersive estimate with $t^{-1}$ decay rate in the case when there is an eigenvalue at zero energy but no resonances.

Full work available at URL: https://arxiv.org/abs/1201.2206

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